Methods Summary

In real-time, dividing deaths-to-date by cases-to-date leads to a biased estimate of the case fatality ratio (CFR), because this calculation does not account for delays from confirmation of a case to death, and under-reporting of cases.
Using the distribution of the delay from hospitalisation-to-death for cases that are fatal, we can estimate how many cases so far are expected to have known outcomes (i.e. death or recovery), and hence adjust the naive estimates of CFR to account for these delays.
The adjusted CFR does not account for under-reporting. However, the best available estimates of CFR (adjusting or controlling for under-reporting) are in the 1% - 1.5% range [1–4]. Large studies in China and South Korea estimating the CFR at 1.38% (95% CrI: 1.23–1.53%)[2] and 1.4% (95% CrI: 1.2-1.7%)[4] respectively. Based on these studies, and for simplicity, we assume a baseline CFR of 1.4% for our analysis.
If a country has an adjusted CFR that is higher (e.g. 20%), it suggests that only a fraction of cases have been reported (in this case, 1.420=7.0% cases reported approximately).

Current estimates for percentage of symptomatic cases reported for countries with greater than ten deaths

Temporal variation

Figure 1: Temporal variation in reporting rate. We calculate the percentage of symptomatic cases reported on each day a country has had more than ten deaths. We then fit a Generalised Additive Model to these data (see Temporal variation model fitting section for details), highlighting the temporal trend of each countries reporting rate. The red shaded region is the 95% CI of fitted GAM.

Current estimates

Figure 2: Plotting the estimates for the proportion of symptomatic cases reported in different countries using cCFR estimates. Blue shading is the 2.5% - 97.5% confidence range.

Table of current estimates

Country	Percentage of symptomatic cases reported (95% CI)	Total cases	Total deaths
Albania	11% (6.9% - 20%)	377	22
Algeria	3.9% (3.1% - 5%)	1423	173
Andorra	17% (10% - 31%)	526	21
Argentina	19% (13% - 27%)	1628	53
Australia	100% (72% - 100%)	5844	42
Austria	45% (35% - 57%)	12297	220
Bangladesh	4.7% (2.7% - 9.5%)	123	12
Belarus	16% (8.5% - 32%)	700	13
Belgium	7.7% (6.5% - 8.9%)	20814	1632
Bolivia	7.8% (4.4% - 15%)	194	14
Bosnia and Herzegovina	15% (9.2% - 24%)	695	28
Brazil	11% (9.1% - 13%)	12056	553
Bulgaria	18% (11% - 32%)	549	22
Burkina Faso	14% (7.8% - 25%)	364	17
Canada	26% (21% - 33%)	16653	323
Chile	73% (47% - 100%)	4815	37
China	33% (29% - 38%)	82698	3335
Colombia	20% (14% - 30%)	1579	46
Croatia	52% (29% - 100%)	1222	16
Cyprus	20% (11% - 40%)	465	14
Czechia	42% (30% - 59%)	4822	78
Democratic Republic of the Congo	5.6% (3.4% - 9.8%)	161	18
Denmark	17% (13% - 21%)	4681	187
Dominican Republic	12% (8.8% - 17%)	1828	86
Ecuador	13% (9.9% - 16%)	3747	191
Egypt	9% (6.6% - 12%)	1073	78
Estonia	42% (24% - 76%)	1108	19
Finland	53% (33% - 89%)	2176	27
France	5.6% (4.9% - 6.4%)	74390	8911
Germany	42% (36% - 49%)	99225	1607
Greece	17% (12% - 24%)	1755	79
Honduras	7.8% (4.8% - 13%)	305	22
Hungary	11% (7.4% - 16%)	817	47
India	16% (12% - 21%)	4421	114
Indonesia	7.3% (5.8% - 9.2%)	2491	209
Iran	12% (11% - 14%)	60500	3739
Iraq	10% (7.4% - 15%)	1031	64
Ireland	18% (14% - 24%)	5364	174
Israel	90% (62% - 100%)	8904	57
Italy	6.7% (5.9% - 7.6%)	132547	16525
Japan	30% (22% - 42%)	3817	80
Lebanon	25% (15% - 46%)	541	19
Lithuania	38% (21% - 77%)	843	14
Luxembourg	52% (35% - 81%)	2843	41
Malaysia	46% (32% - 67%)	3793	62
Mexico	9.7% (7.3% - 13%)	2439	125
Moldova	21% (12% - 38%)	965	19
Morocco	7.1% (5.2% - 9.8%)	1120	80
Netherlands	6.7% (5.7% - 7.7%)	18803	1867
North Macedonia	15% (9.2% - 27%)	570	21
Norway	79% (55% - 100%)	5755	59
Pakistan	36% (25% - 53%)	3864	54
Panama	22% (15% - 32%)	2100	55
Peru	13% (9.2% - 17%)	2561	92
Philippines	12% (9.3% - 16%)	3660	163
Poland	22% (16% - 30%)	4413	107
Portugal	23% (18% - 28%)	11730	311
Puerto Rico	10% (6.2% - 18%)	513	21
Romania	14% (11% - 18%)	4057	157
Russia	53% (36% - 80%)	6343	47
San Marino	8.2% (5.4% - 13%)	277	32
Saudi Arabia	41% (27% - 64%)	2523	38
Serbia	16% (11% - 24%)	2200	58
Slovenia	28% (18% - 46%)	1021	30
South Africa	100% (55% - 100%)	1686	12
South Korea	64% (50% - 83%)	10331	192
Spain	7.1% (6.2% - 8%)	135032	13055
Sweden	10% (8.2% - 12%)	7206	477
Switzerland	28% (23% - 34%)	21574	584
Thailand	64% (39% - 100%)	2258	26
Tunisia	16% (9.4% - 27%)	574	22
Turkey	20% (17% - 24%)	30217	649
Ukraine	16% (10% - 24%)	1319	38
United Arab Emirates	70% (35% - 100%)	2076	11
United Kingdom	4.9% (4.2% - 5.5%)	51608	5373
United States of America	17% (15% - 19%)	368196	10989

Table 1: Estimates for the proportion of symptomatic cases reported in different countries using cCFR estimates based on case and death timeseries data from the ECDC. Total cases and deaths in each country is also shown. Confidence intervals calculated using an exact binomial test with 95% significance.

Adjusting for outcome delay in CFR estimates

During an outbreak, the naive CFR (nCFR), i.e. the ratio of reported deaths date to reported cases to date, will underestimate the true CFR because the outcome (recovery or death) is not known for all cases [5]. We can therefore estimate the true denominator for the CFR (i.e. the number of cases with known outcomes) by accounting for the delay from confirmation-to-death [1].

We assumed the delay from confirmation-to-death followed the same distribution as estimated hospitalisation-to-death, based on data from the COVID-19 outbreak in Wuhan, China, between the 17th December 2019 and the 22th January 2020, accounting right-censoring in the data as a result of as-yet-unknown disease outcomes (Figure 1, panels A and B in [7]). The distribution used is a Lognormal fit, has a mean delay of 13 days and a standard deviation of 12.7 days [7].

To correct the CFR, we use the case and death incidence data to estimate the proportion of cases with known outcomes [1,6]:

\[ u_{t} = \frac{ \sum_{j = 0}^{t} c_{t-j} f_j}{c_t}, \]

where \(u_t\) represents the underestimation of the proportion of cases with known outcomes [1,5,6] and is used to scale the value of the cumulative number of cases in the denominator in the calculation of the cCFR, \(c_{t}\) is the daily case incidence at time, \(t\) and \(f_t\) is the proportion of cases with delay of \(t\) between confirmation and death.

Approximating the proportion of symptomatic cases reported

At this stage, raw estimates of the CFR of COVID-19 correcting for delay to outcome, but not under-reporting, have been calculated. These estimates range between 1% and 1.5% [1–3]. We assume a CFR of 1.4% (95% CrI: 1.2-1.7%), taken from a recent large study [3], as a baseline CFR. We use it to approximate the potential level of under-reporting in each country. Specifically, we perform the calculation \(\frac{1.4\%}{\text{cCFR}}\) of each country to estimate an approximate fraction of cases reported.

Temporal variation model fitting

We estimate the level of under-reporting on every day for each country that has had more than ten deaths. We then fit a Generalised Additive Model (GAM) of the form \[ \mathbb{E}[\log(D)] = \beta_0 + \beta_1 x_1 + ... + \beta_p x_p,\] specifying a Poisson distribution on deaths (D) as the response variable. The model has a log-link function and a log-offset (\(\kappa\)) consisting of the daily known-outcomes \(u_t\) and the cCFR estimate for that country on that day \(\text{cfr}_t\). The model can then be written as \[ D \sim s(t) + \underbrace{\log(u_t c_t) + \log(\text{cfr}_t)}_{:=log(κ)} \] where \(s(t)\) is a smoothing spline, fitted through the time points (days) for which we have data.

Limitations

Implicit in assuming that the under-reporting is \(\frac{1.4\%}{\text{cCFR}}\) for a given country is that the deviation away from the assumed 1.4% CFR is entirely down to under-reporting. In reality, burden on healthcare system is a likely contributing factor to higher than 1.4% CFR estimates, along with many other country specific factors.

The following is a list of the other prominent assumptions made in our analysis:

We assume that people get tested upon hospitalisation. A few examples where this is not the case are Germany and South Korea, where people can get tested earlier.
We assume that hospitalisation to death from early Wuhan is representative of the all the other countries (by using the distribution parameterised using early Wuhan data) and that all countries have the same risk and age profile as Wuhan.
Severity of COVID-19 is known to increase with age. Therefore, countries with older populations will naturally see higher death rates. We are extending this analysis to adjust for the age distribution for countries with more than five reported deaths and where age distribution data is available.
All results are linked and biased by the baseline CFR, assumed at 1.4% [3].
The under-reporting estimate is sensitive to the baseline CFR, meaning that small errors in it lead to large errors in the estimate for under-reporting.
There are several sources of uncertainty in this analysis: the reported mean and SD of the delay distribution; the cCFR for each country (on each day) and the baseline CFR. We use the lower and upper mean and SD of the delay distribution to produce the widest interval. The other two uncertainties are just carried through directly in the subsequent calculations, which is crude. An ongoing extension is developing this into a fully Bayesian model, which deals with these different uncertainties more rigorously.

Code and data availability

The code is publically available at https://github.com/thimotei/CFR_calculation. The data required for this analysis is a time-series for both cases and deaths, along with the corresponding delay distribution. We scrape this data from ECDC, using the NCoVUtils package [8].

References

1 Russell TW, Hellewell J, Jarvis CI et al. Estimating the infection and case fatality ratio for covid-19 using age-adjusted data from the outbreak on the diamond princess cruise ship. medRxiv 2020.

2 Verity R, Okell LC, Dorigatti I et al. Estimates of the severity of covid-19 disease. medRxiv 2020.

3 Guan W-j, Ni Z-y, Hu Y et al. Clinical characteristics of coronavirus disease 2019 in china. New England Journal of Medicine 2020.

4 Shim E, Mizumoto K, Choi W et al. Estimating the risk of covid-19 death during the course of the outbreak in korea, february-march, 2020. medRxiv 2020.

5 Kucharski AJ, Edmunds WJ. Case fatality rate for ebola virus disease in west africa. The Lancet 2014;384:1260.

6 Nishiura H, Klinkenberg D, Roberts M et al. Early epidemiological assessment of the virulence of emerging infectious diseases: A case study of an influenza pandemic. PLoS One 2009;4.

7 Linton NM, Kobayashi T, Yang Y et al. Incubation period and other epidemiological characteristics of 2019 novel coronavirus infections with right truncation: A statistical analysis of publicly available case data. Journal of Clinical Medicine 2020;9:538.

8 Abbott S MJ Hellewell J. NCoVUtils: Utility functions for the 2019-ncov outbreak. doi:105281/zenodo3635417 2020.

Aim